Decomposition Laws

Abstract

In this chapter we describe the properties and the significance of decomposition laws in extensions of algebraic number fields. In this way we settle a basis for the formulation of these laws, and we mention results from abelian and Kummer extensions which offer important contributions to the investigations below.

In the first section we define basic notions and name characteristic quantities appearing in connection with decomposition laws. We explain how the decomposition of a prime ideal can be obtained by using the product decomposition of a polynomial into irreducible factors (Theorem 2.1.1) and how the decomposition behaviour in general extensions can be read off from suited Galois extensions (Theorem 2.1.2). We emphasize the simplification of the decomposition law in the case of Galois extensions.